Number Theory

    

A Conservative Structural Argument Toward the Infinitude of Twin Primes via Inclusion-Exclusion

Authors: Younghwan Yun

We propose a conservative structural framework to address the twin prime conjecture, aiming to demonstrate the unavoidable recurrence of twin prime pairs across the number line. By systematically applying the inclusion-exclusion principle within bounded intervals [p2 n−1, p2n), where pn is the n-th prime, we estimate the minimal lower bound of surviving 6k ± 1 pairs after sieving out all multiples of smaller primes. Our analysis shows that any composite survivor within such intervals would require a prime factor at least as large as pn, leading to a contradiction by exceeding the interval’s up-per bound. We derive an explicit minimal estimate Tn for the number of twin primepairs and show that it grows unboundedly with n. This non-probabilistic approach provides a concrete methodological pathway suggesting that the periodic sieve structure necessarily sustains infinitely many twin prime pairs, offering strong structural support for the twin prime conjecture.

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[v1] 2025-07-14 20:41:24

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